Page:Scientific Papers of Josiah Willard Gibbs.djvu/304

268 where   It will be observed that $$A$$ vanishes when the composition of the two homogeneous masses is identical, while $$B$$ and $$C$$ do not, in general, and that the value of $$A$$ is negative or positive according as the mass specified by $$'$$ contains the component specified by $$_{1}$$ in a greater or less proportion than the other mass. Hence, the values both of $$\left( \frac{d\sigma}{dt} \right)_{p}$$ and of $$\left( \frac{d\sigma}{dp} \right)_{t}$$ become infinite when the difference in the composition of the masses vanishes, and change sign when the greater proportion of a component passes from one mass to the other. This might be inferred from the statements on page 99 respecting coexistent phases which are identical in composition, from which it appears that when two coexistent phases have nearly the same composition, a small variation of the temperature or pressure of the coexistent phases will cause a relatively very great variation in the composition of the phases. The same relations are indicated by the graphical method represented in figure 6 on page 125.

With regard to gas-mixtures which conform to Dalton's law, we shall only consider the fundamental equation for plane surfaces, and shall suppose that there is not more than one component in the liquid which does not appear in the gas-mixture. We have already seen that in limiting the fundamental equation to plane surfaces we can get rid of one potential by choosing such a dividing surface that the superficial density of one of the components vanishes. Let this be done with respect to the component peculiar to the liquid, if such there is; if there is no such component, let it be done with respect to one of the gaseous components. Let the remaining potentials be eliminated by means of the fundamental equations of the simple gases. We may thus obtain an equation between the superficial tension, the temperature, and the several pressures of the simple gases in the gas-mixture or all but one of these pressures. Now, if we eliminate $$d\mu_{2}, d\mu_{3}$$, etc. from the equations where the suffix $$_{1}$$ relates to the component of which the surface-density has been made to vanish, and $$\gamma_{2}, \gamma_{3}$$, etc. denote the densities